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Steiner system
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' ≥ 2.
A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design.
This definition is relatively modern, generalizing the ''classical'' definition of Steiner systems which in addition required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') was called a ''Steiner quadruple system'', and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
A long-standing problem in design theory is if any nontrivial (''t'' < ''k'' < ''n'') Steiner systems have ''t'' ≥ 6; also if infinitely many have ''t'' = 4 or 5. This was claimed to be solved in the affirmative by Peter Keevash.
== Examples ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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